Grasping Force Estimation by sEMG Signals and Arm Posture:Tensor Decomposition Approach

Sanghyun Kima, Joowan Kima, Mingon Kima, Seungyeon Kima, Jaeheung Parka,b,∗

aDepartment of Transdisciplinary Studies, Seoul National University, Suwon, 16229, Republic of Korea.bAdvanced Institute of Convergence Science and Technology, Suwon, 16229, Republic of Korea.

Abstract

Grasping force estimation using surface Electromyography (sEMG) has been actively investigated as it can increasethe manipulability and dexterity of prosthetic hands and robotic hands. Most of the current studies in this area onlyfocus on finding the relationship between sEMG signals and the grasping force without considering the arm posture.Therefore, regression models are not suitable to predict grip forces in various arm postures. In this paper, a methodto predict the grasping force from sEMG signals and various grasping postures is developed. The proposed algorithmuses a tensor algebra to train a multi-factor model relevant to sEMG signals corresponding to various grasping forcesand postures of the wrist and forearm in multiple dimensions. The multi-factor model is then decomposed into the fourindependent fac-tor spaces of the grasping force, sEMG signals, wrist posture, and forearm posture. Moreover, when aparticipant exe-cutes a new posture, new factors for the wrist and forearm are interpolated in the factor spaces. Thus,the grasping force with various postures can be predicted by combining these factors. The effectiveness of the proposedmethod is verified through experiments with ten healthy subjects demonstrating higher performance of the graspingforce prediction than the previous algorithm

Keywords: Surface Electromyography (sEMG), Grasping Force Estimation, Tensor Decomposition, Multi-factor Model

1. Introduction

A surface Electromyographic (sEMG) signal is a bio-logical electrical activity produced by muscle contraction.As sEMG signals are measured from the surface above themuscle on the skin, they can be used for a variety appli-cations including diagnostic tools to detect muscle fatigue[1], myoelectric controls for prostheses [2, 3], and human-machine interfaces in virtual reality by estimating the up-per arm and finger motion [4, 5, 6].

In particular, the grasping force estimation has beenactively studied for the past decades it can provide a greatdeal of manipulation capabilities to prosthetic hands androbotic hands. Since Arm-strong et al. studied a method-ology to predict hand forces from sEMG signals using a lin-ear regression [7], various regression models between multi-channel sEMG signals and forces have been proposed usingthe Principal Component Analysis (PCA) [8], IndependentComponent Analysis (ICA) [9], Sample Entropy Method(SEM) [10], Support Vector Machine (SVM) [11, 12, 13],Artificial Neural Network (ANN) [14, 15], Extreme Learn-

IThis work was supported by Industrial strategic technology de-velopment program (No. 10077538 ) funded by the Ministry ofTrade, Industry & Energy (MI, Korea).

∗Corresponding authorEmail address: park73@snu.ac.kr (Jaeheung Park)URL: dyros.snu.ac.kr (Jaeheung Park)

ing Machine (ELM) [16, 17], and the modified logarithmicregression [18].

These approaches, however, present a limitation: therelationship between the sEMG signals and the grip forcewas proposed without considering the arm posture suchas the wrist and forearm angle although the posture canaffect the sEMG signals. In fact, as the contraction stateof the muscles depends on the posture of the arm andthe pattern of the sEMG signals changes [19, 20]. Conse-quently, the results of the predicted grip force using theseregression models present large errors for postures otherthan a neutral arm posture.

To overcome this limitation, there are several stud-ies about the correlation between the grasping force andsEMG signals with the posture of the arm. Duque et al.investigated the regression model taking into account theposture of the wrist (flexion and extension, see Fig. 1(a))by using the multiple logarithmic regression method [21].In another study, Sidek and Mohideen researched the ANNmodel reflecting sEMG signals and the wrist’s posture [22].However, these studies still have a limitation because theydid not take into account the posture of the forearm.

On the other hand, Keir and Mogk presented the equa-tion for the prediction of the grip force using sEMG signalsas well as the posture of the wrist and forearm (supinationand pronation, see Fig. 1(b)) by using the second orderpolynomial regression [23]. However, the estimated force

Preprint submitted to Journal of Bionic Engineering March 14, 2019

Hypertension

Flexion

Extension

(a)

Supination

Pronation

(b)

Figure 1. Posture of arm: (a) posture of wrist: flexion, extension,and hypertension, (b) posture of forearm: supination and pronation.

with this equation sometimes shows erroneous results be-cause the inputs of the equation are discrete variables. Forexample, the wrist posture was input as discrete data (ex-tension=1, neutral=2, flexion=3) in their equation model.Consequently, to the best of our knowledge, there is nosuitable regression model between sEMG signals and gripforce with various postures of the arm.

Therefore, the following questions are raised: if thepattern of the sEMG signals is defined as the combinationof factors reflecting the grasping force and the posture ofthe arm, can we build a multi-factor model to train thesEMG signals? In addition, can we extract these factorsindependently in the multi-factor model? Finally, can wepredict the grasping force with the untrained arm postureby using these factors? These issues constitute the mainfocus of this study.

Herein, a new type of regression model is proposedto answer the aforementioned questions. The aim of thisstudy is to accurately predict the grasping force by con-sidering the postures of the arm. The proposed algorithmuses a tensor composed of data relevant to the sEMGsignals corresponding to the various grasping forces andposture of the wrist and forearm in multiple dimensions.This tensor, which is regarded as a multi-factor model inthis paper, is then decomposed into the four independentfactor spaces of information of the grasping force, sEMGsignals, wrist and forearm posture by using the tensor de-composition method. Furthermore, if the user has a newarm posture that is not included in the multi-factor model,new factors related on the wrist and forearm posture areobtained by considering similarities with the training pos-tures in the multi-factor model. Thus, the estimated forcewith any posture can be generated by combining these in-terpolated factors.

This paper is organized as follows. First, we presenthow to constitute the multi-factor model in the tensorspace and estimate the grasping force using the multi-factor model in Sec. 2. Section 3 presents the experi-mental setup and data collections and Section 4 shows thevalidation process for the proposed algorithm. Section 5describes the experimental results and discussion of theproposed algorithm by comparing to other method andthe paper is concluded in Sec. 6.

To enhance readability, Table 1 denotes the symbolsand their corresponding meanings in this paper.

Table 1. Notation and symbols

Symbol Descriptiona scalar (lowercase letters)a vector (bold lowercase letters)A matrix (bold uppercase letters)A tensor (underlined bold upper letters)

An mode-n unfoldingaijk i, j, kth element of tensor A×n mode-n multiplication

2. Technical Approach

The proposed algorithm for estimating grasping forcesusing tensor algebra consists of two parts: the trainingpart to construct the multi-factor model and the graspingforce estimator for the probe data.

In the following subsection, we briefly review a basictensor algebra and the tensor decomposition technique.Then, in Sec. 2.2, we introduce how to construct the multi-factor model for the grasping estimation and the methodused to extract four factor spaces reflecting the characteris-tics of the grasping force, sEMG patterns, the wrist’s pos-ture, and the forearm’s posture in the multi-factor model.The multi-factor model based on the Tucker decompositionis applied to separate these factors independently. In Sec.2.3, we discuss the approaches to calculate the factors cor-responding to an untrained arm posture and to estimatethe grasping force of the user with l2-norm minimization,respectively.

2.1. Mathematical Background: Tensor AlgebraThis section is a condensation of [24] and [25]. A ten-

sor, which is a multi-way array, provides a compact rep-resentation for multi-dimensional data. Therefore, tensoralgebra can be used to analyze the multi-way signal pat-tern such as walking motion [26], hand motion [27], andEEG data [28].

Let us define X ∈ Ri1×i2×···×in is n-th order tensor.Thus, a vector is a first-order tensor, and a matrix is asecond-order tensor. A tensor can be expressed as a ma-trix to use matrix operations. This process is referred toas a tensor unfolding. For example, let X be a tensorof order 3 with X ∈ Ri1×i2×i3 . Then, the first mode un-folding of X is written X1, and its dimension is Ri1×i2i3 .Likewise, dimensions of mode-2 and mode-3 unfolding areX2 ∈Ri2×i1i3 and X3 ∈Ri3×i1i2 , respectively, as shown inFig. 2(a). A refolding which is transformed from a matrixto a tensor reverses unfolding operation.

For multiplication of a tensor and a matrix (or a vec-tor), mode-n multiplication is defined. For instance, mode-2 multiplication of a third order tensor X ∈Ri1×i2×i3 anda matrix B ∈ Rj2×i2 is denoted by

Y = X×2 B, (1)

2

Mode 1

unfolding𝑿𝟏

𝒊𝟏

𝒊𝟑𝒊𝟐

𝒊𝟐

𝒊𝟏

𝒊𝟑

𝑿𝟐𝒊𝟏

𝒊𝟑

𝒊𝟐

𝒊𝟑

𝒊𝟏

𝑿𝟑𝒊𝟏

𝒊𝟑

𝒊𝟐

𝒊𝟏

𝒊𝟐

Mode 2

unfolding

Mode 3

unfolding

𝒊𝟐

𝒊𝟑

(a)

= +𝒀 𝑬𝑮

𝑼𝟏

𝑼𝟑

𝑼𝟐

(b)

Figure 2. Illustration of tensor representation and operations: (a)mode-n unfolding of third order tensor, (b) Tucker decomposition ofthird order tensor.

where dimension of a tensor Y is Ri1×j2×i3 . Alternatively,it can also be expressed by using the unfolding operation,as follow:

Y2 = B×X2. (2)

Now, let us consider the decomposition method for ten-sor algebra. In matrix algebra, Singular Vector Decom-position (SVD) is a representative factorization by usingbasis vectors (singular vectors) of a matrix. An SVD of amatrix X ∈ Ri1×i2 is expressed as

X = UΣV T , (3)

where U ∈ Ri1×i1 , V ∈ Ri2×i2 , and Σ ∈ Ri1×i2 are theleft and right singular matrix, and diagonal matrix, whichconstitutes singular values, respectively.

Similarly, in the tensor space, the Tucker decomposi-tion [29], which is regarded as High Order SVD (HOSVD),decomposes an n-th order tensor Y ∈ Ri1×i2···×in intoa core tensor G ∈ Rj1×j2···×jn and thin factor matricesUn ∈ Rjn×in ,

Y = G×1 U1×2 U2...×n Un +E, (4)

where E ∈Ri1×i2···×in is the error tensor by the factoriza-tion and the number of rows of n-th factor matrix, jn, isless than or equal to the number of columns, in.

The column vectors of Un are the basis vectors of themode-n unfolding matrix, Yn. Moreover, like Σ of (3),the core tensor, G, captures the relationship among factormatrices Un. Figure 2(b) illustrates the Tucker methodof a third order tensor.

2.2. Multi-factor ModelAs aforementioned Sec. 1, the pattern of sEMG sig-

nals is obviously influenced by multi-factors including themagnitude of grasping force and the posture of wrist andforearm. Thus, unlike previous studies that constructedthe single-factor model for examining the relationship be-tween a grip strength and sEMG signals [9-19], our pro-posed framework uses the multi-factor model using thetensor algebra.

For this reason, as training data for our proposed algo-rithm, the sEMG signals with increasing the grasping forceat several wrist and forearm angles are recorded. Thistraining data set constitutes a fourth order tensor for thegallery in the proposed algorithm. In our experiments, thedimensions of the tensor for the gallery Y ∈ Ri1×i2×i3×i4

indicate the overall number of sample grasping force (i1),the number of sEMG signal (i2), the number of the wrist’angles (i3), and the number of the forearm’s angles (i4).Then, as mentioned in Sec. 2.1, the gallery tensor is de-composed using the Tucker method,

Y = G×1 U1×2 U2×3 U3×4 U4 +E, (5)

where G ∈ Rj1×j2×j3×j4 is the core tensor, Uk ∈ Rjk×ik

denotes the factor matrices of mode-k, and E ∈Ri1×i2×i3×i4

is the error tensor, respectively.The core tensor, G, and each factor matrix, Uk, are

calculated by the Alternative Least Square (ALS) algo-rithm [30] to minimize the Frobenius norm, as follow:

minimizeG,U1,U2,U3,U4

‖Y −G×1 U1×2 U2×3 U3×4 U4‖2F

subject to G ∈ Rj1×j2×j3×j4 ,

Uk ∈ Rik×jk : column−orthogonal.(6)

In (6), each factor matrix ,Uk, is an independent matrix toeach other and it contains the intrinsic factors, uk,i ∈Rjk

and it is expressed as

Uk = [uk,1, · · · ,uk,ik ]. (7)

Similar to SVD, the factor matrix, U1, spans the spaceof the grasping force and the factor matrix of sEMG sig-nals, U2, represents the principal components of the sEMGpattern regardless of the grasping force and the arm pos-ture. The factor matrices U3 and U4 span the space ofthe wrist and forearm posture, respectively. Especially,the factor matrices of the wrist, U3, and the forearm pos-ture, U4, contain the intrinsic posture parameters that areinvariant to the patterns of sEMG signals and the graspingstrength.

3

Gallery Tensor

Training

sEMG Signals Multi-Factor Model Generation Using Tucker Decomposition

Grip Force

Wrist Angle

Forearm Angle

Factors Interpolation for Arm Postures

sEMG Signals L2 optimization for Estimating Grip Force

Wrist Angle

Forearm Angle

𝒀 𝑮 ×𝟏 𝑼𝟏 ×𝟐 𝑼𝟐 ×𝟑 𝑼𝟑 ×𝟒 𝑼𝟒

Gallery Tensor Feature Space for Force

Feature Space for EMG Pattern

Feature Space for Wrist

Feature Space for Forearm

𝑼𝟑 𝑼𝟒 𝒑𝟑 𝒑𝟒C-spline 𝒚 𝑮

EMG Signals

×𝟏 𝒑𝟏×𝟐 𝑼𝟐 𝒑𝟑×𝟑 ×𝟒 𝒑𝟒

Estimated Force

(a)

Gallery Tensor

Training

sEMG Signals Multi-Factor Model Generation Using Tucker Decomposition

Grip Force

Wrist Angle

Forearm Angle

Factors Interpolation for Arm Postures

sEMG Signals L2 optimization for Estimating Grip Force

Wrist Angle

Forearm Angle

𝒀 𝑮 ×𝟏 𝐔𝟏 ×𝟐 𝑼𝟐 ×𝟑 𝑼𝟑 ×𝟒 𝑼𝟒

Gallery Tensor Feature Space for Force

Feature Space for EMG Pattern

Feature Space for Wrist

Feature Space for Forearm

𝑼𝟑 𝑼𝟒 𝒑𝟑 𝒑𝟒C-spline 𝒚 𝑮

EMG Signals

×𝟏 𝒑𝟏×𝟐 𝑼𝟐 𝒑𝟑×𝟑 ×𝟒 𝒑𝟒

Estimated Force

(b)

Figure 3. Schematic diagram for grasping force estimation: (a) multi-factor model for training data, (b) grasping force estimation for probedata.

Finally, using mode-2 multiplication, (5) can be re-shaped so that it represents the decomposition of the onlyU1, U3, and U4 by means of

Y ' G×1 U1×3 U3×4 U4, (8)

where the new core tensor, G = G×2 U2. The dimen-sions of the tensor are G ∈ Rj1×i2×j3×j4 . Consequently,through (8), the proposed multi-factor model can be rep-resented by a combination of factor spaces for the graspingforce (U1), wrist posture (U3), and forearm posture (U4).

Given the multi-factor model, the trained sEMG signal(y ∈ Ri2) in the multi-factor model can be written as,

y = G×1 u1,i×3 u3,j×4 u4,k, (9)

where u1,i, u3,j , and u4,k are the corresponding factorsof the magnitude of the grasping force, the wrist posture,and the forearm posture, respectively.

2.3. Grasping Force EstimatorThanks to the multi-factor model, the relation between

four independent factor spaces (grasping force, sEMG sig-nals, wrist posture, and forearm posture) can be described.Using these factor spaces, in this section, we describe howto predict the grasping force with probe data. As shown inFig. 3(b), the procedure for the grasping force estimationusing the multi-factor model consists of two steps. First,when a user takes an arbitrary arm posture, the corre-sponding factors are interpolated in the factor spaces of thewrist posture (U3), and forearm posture (U4). With thesefactors and the multi-factor model, our system predicts therelationship between sEMG signals and grip strength inany arm posture. Second, the grasping force is estimated

3,2u3,1u

3,3u

3,4u3,5u

3p

Figure 4. Illustration of factor matrix for wrist posture (U3) andestimated wrist factor (p3) in three-dimensional space. The blackline is generated by the periodic interpolating cubic spline curve.

by l2-norm minimization with the actual sEMG signals ofthe user and the interpolated factors.

In the following subsections, we describe the details ofthe proposed grasping force estimation method using themulti-factor model.

2.3.1. Factor Interpolation for Arm PostureThe corresponding factors of the wrist, p3 ∈ Rj3 , and

forearm, p4 ∈Rj4 , with respect to an arbitrary posture canbe generated by the factor spaces for the wrist (U3) andforearm (U4) in the multi-factor model. To find these cor-responding factors, we generate an interpolation curve thatconnects each factor of trained factor spaces. Then, thecorresponding factor is selected as a point on the interpo-lation curve according to the angle difference between thetrained postures and the arbitrary posture. To generate

4

Gripper(w. Force Sensor)

Rotation Platefor Forearm

Arm Rest

Rotation Platefor Wrist

(a) (b)

Supination

Pronation

Hypertension Flexion

-30 +30

+90

-90

Wrist

Fore

arm

(c)

Figure 5. Experimental system overview: (a) experimental kit to maintain arm posture, (b) experimental setup with sEMG devices, (c)measurement range of arm posture during experiments.

the interpolation curve, we used the well-known periodicinterpolating cubic spline curve [31, 32].

Fig. 4 shows an illustration of how to select the wristfactor with respect to the untrained wrist posture in thefactor matrix of the wrist, U3. Let us assume that themulti-factor model is constructed by performing five wristpostures (-30◦, -15◦, 0◦, 15◦, and 30◦). Then, the wristfactors in the multi-factor model (u3,1, u3,2, u3,3, u3,4,and u3,5) indicate the five trained wrist posture. Withthese trained factors of the wrist, if the user takes 7.5◦

wrist flexion, the corresponding wrist factor, p3. couldbe selected as the midpoint between u3,2 and u3,3 in thecubic spline curve.

2.3.2. Grasping Force EstimationAs the p3 and p4 are selected in factor matrices for

the wrist and forearm posture in the multi-factor model,the corresponding factor of the grasping force, p1 ∈ Rj1 ,can be calculated in factor matrix for grasping force, U1.Given corresponding posture factors with respect to arbi-trary posture, untrained sEMG signals (ym ∈ Ri2), andthe trained multi-factor model, the corresponding factorof the grasping force with an untrained posture, p1, is es-timated by l2-norm minimization, as follow:

minimizep1,t

‖ym− G×1 p1×3 p3×4 p4‖2

subject to p1 = tu1,i + (1− t)u1,i+1,

0≤ t < 1.

(10)

Here, t is an interpolating coefficient between 0 and 1.Thus, p1 is a linear interpolation value in the factor spaceof the grasping force, U1.

By obtaining the corresponding factor of the graspingforce, p1, and the interpolating coefficient, t, by (10), thegrasping force, fp, can be estimated as follow:

fp = tf i + (1− t)f i+1 (11)

where f i and f i+1 indicate the magnitude of the trainedgrasping forces of u1,i and u1,i+1, respectively. Conse-quently, the grasping force of probe data is predicted byconsidering the multi-factor including sEMG signals, wrist,and forearm posture.

3. Experimental Setup

The proposed algorithm was verified through experi-ments conducted with ten subjects (five males and fivefemales). The experimental protocol was approved by theInstitutional Review Board of Seoul National University(No. SNU 17-05-039 ). In this experiment, to validatethe generality of the proposed model, the dataset was col-lected over two days. The datasets of Day 1 and Day 2were used to train the multi-factor model and validate theproposed algorithm, respectively. The following subsec-tions describe the detail of the system configuration anddataset.

3.1. System OverviewThe experimental setup for measuring the grip force

and sEMG signals is described in Fig. 5(a) and 5(b). Ourexperimental kit was designed using two rotation plates toassist the subject’s arm postures. Thus, the user can easilymaintain the posture of the arm in the training process ofmeasuring force and sEMG signals. When the user gripsthe rod, the grasping force is tracked by a force sensor(ATI, Mini-45, Resolution: 0.04N). In addition, the wire-less sEMG measurement system (Trigo, Delsys) is used torecord the sEMG signals. Both force and sEMG data aresampled at a rate of 500 Hz and synchronized in time bythe single computer with quad-core 3.4 GHz and 12 GBRAM.

3.2. Dataset (Day 1) for Multi-factor ModelOur experimental schemes to construct the multi-factor

model is summarized as follows.

5

0 5 10Time (sec)

0

10

20

30

40

For

ce (

N)

(a)

0 5 10Time (sec)

0

10

20

30

40

For

ce (

N)

(b)

0 5 10Time (sec)

0

10

20

30

40

For

ce (

N)

(c)

Figure 6. Sequence for time-aligned signals: (a) example of the raw grasping force signals (10 samples), (b) the resampled force signals, (c)the time aligned signals using DTW.

• sEMG Signals: According to the human anatomy,six major muscles in the forearm are selected to mea-sure sEMG signals[33]: Flexor Carpi Radialis (FCR),Flexor Carpi Ulnaris (FCU), Flexor Digitorum Su-perficialis (FDS), Extensor Carpi Radialis (ECR),Extensor Carpi Ulnaris (ECU), and Extensor Digito-rum Communis (EDC). Thus, the number of sEMGsignals (i2) for the gallery tensor is 6. Also, in orderto reduce the influence of noise, the sEMG signalsfor the multi-factor model were preprocessed usingthe Mean Absolute Value (MAV). Then, the recti-fied sEMG signals by MAV were passed through thezero-phase low pass filter (4th order, 0.5 Hz) to min-imize delay time due to filtering. Note that zero-phase filtering helps preserve features of a filteredtime waveform exactly which occur in the unfilteredsignal.

• Grasping Force: During power grasping, the par-ticipants increase the grasping force linearly for 10seconds from 0 N to 40 N for male and 30 N forfemale. To reduce the effect of outliers, the partic-ipants performed this power grasping three times.Thus, the number of sample grasping force (i1) is15000 (=3 trials × 10 sec × 500Hz), because of oursampling frequency is 500 Hz.

• Arm Posture: As shown in Fig. 5(c), participantscarry out experiments with three wrist position (-30◦, 0◦, and 30◦) and three forearm position( -90◦,0◦, and 90◦) to train the multi-factor model. There-fore, each participant performs a total of 9 measure-ments which is a combination of three wrist (i3) andthree forearm postures (i4). Finally, to minimizemuscle fatigue, there is a 3-minute break betweeneach measurement.

Note that the grasping data of each subject is filteredby using Dynamic Time Warping (DTW) which is a well-

known trajectory matching algorithm by finding an opti-mal alignment between time sequences [34, 35]. It is be-cause the participant cannot maintain his or her rhythmconstantly, as shown in Fig. 6(a).

Based on the DTW algorithm, in our experiments, thefollowing sequences were conducted to align two or moresignals.

1. Align the length of each signal by extending andstretching signals, as shown in Fig. 6(b).

2. Set a reference template for DTW as an average ofevery signal.

3. Apply DTW with the reference template and eachsignal. The final aligned signals are shown in Fig.6(c).

Consequently , the dimensions of the gallery tensor foreach subject Y were R15000×6×3×3 in our experiments.Then, we performed the Tucker decomposition for the multi-factor model of (5) using MATLAB Tensor Toolbox [36].The dimensions of the core tensor (G ∈Rj1×j2×j3×j4) arechosen to include 99 % of the sum of singular values of thecorresponding mode unfolding, in order to avoid overfit-ting.

3.3. Dataset (Day 2) for ValidationTo examine the performance of our proposed algorithm

in various arm postures, we performed four tests, as fol-lows. In each test, participants can increase or decreasethe grasping force freely, within the range of 0N to 40N formale and 30N for female. Five validations were measuredfor each case. The preprocess of sEMG signals is almostthe same as the method explained in Sec. 4.2. The onlydifference is that a real-time low pass filter was appliedto the validation data instead of the zero-phase low-passfilter.

• CASE I: Case I is an experiment to validate the per-formance of the proposed algorithm when the par-ticipants maintain trained arm posture. To do this,

6

each participant maintained a trial in one of ninepostures which are a combination of three wrist pos-tures (-30◦, 0◦, 30◦) and three forearm postures (-45◦, 0◦, 45◦). Therefore, in this case, we can usethe factors of the posture in the multi-factor model,without the interpolation process in Sec. 2.3.1.

• CASE II: Case II is an experiment for verifying howmuch the grip force can be estimated by interpolat-ing the factor of the wrist (p3). The participantsmaintained one of four untrained wrist postures (-15◦, 15◦) with maintaining the trained forearm an-gle (-45◦, 0◦, 45◦). Thus, we can use the factor ofthe forearm in the multi-factor model directly butshould estimate the factor of the wrist (p3).

• CASE III: Similar to Case II, the participants tookone of two untrained forearm postures (-45◦, 45◦)with maintaining the trained wrist angle (-30◦, 0◦,30◦) in Case III. Thus, the factor of the forearm (p4)should be estimated by the interpolation.

• CASE IV: Case IV is an experiment that predictsthe grasping force with untrained wrist (p3) andforearm postures (p4). Each participant performedone of four measurements which is a combination oftwo wrist postures (-15◦, 15◦) and two forearm pos-tures (-45◦, 45◦).

4. Validation Process

In this section, we describe the validation indexes andthe comparison algorithm for validating the performanceof the proposed algorithm.

4.1. Validation ParametersFor validating each test, Normalized Root Mean Squared

Error (NRMSE) and Coefficient of Correlation (CORR)were used to show the error of our proposed regressionmodel. For the measured force in probe data, each valueis calculated as follows:

NRMSE =

√∑(fm,i−fp,i)2

N

fm,max−fm,min(12)

and

CORR= N∑

fm,ifp,i−∑

fm,i∑

fp,i√N

∑fm,i

2−(∑

fm,i)2√

N∑

fp,i2−(

∑fp,i)2

,

(13)where

∑is the simplified form of

∑Ni=1. In (12) and (13),

fm,i and fp,i represent the i-th observed data (the mea-sured force) and the i-th regression data (the estimatedforce). Also fm,max and fm,min indicate the maximumand minimum value of the observed data set, respectively.N is the number of the observed data set.

Thus, the smaller NRMSE represents the smaller differ-ence error between the estimated force and the measuredforce at each time. Also, the closer the CORR is to 1, thestronger linear relationship between the estimated forceand the measured force.

4.2. Regression Model for ComparisonBecause there is no proper regression model for esti-

mating the grip force while considering the posture of thewrist and the forearm, we decided to modify the multipleregression model of Keir and Mogk [23] for comparison.The modified regression model for comparison (hereafterreferred to as modified Keir’s method) is represented asfollow:

fm =6∑

j=1(ajym,j + bjym,j

2) +a7pw + b7pf + c, (14)

where ym,j , pw, and pf indicate an amplitude of j-thsEMG signal (mV), the wrist’s angle (rad), and the fore-arm’s angle (rad), respectively. aj and bj represent thefirst and second order coefficients of the regression model.a7, b7, and c are coefficients for the wrist and forearm anda constant offset.

5. Results and Discussion

Figs. 7 - 10 summarize the validation results of the pro-posed algorithm and the modified Keir’s method, respec-tively. In Case I, the average NRMSE and CORR of theproposed algorithm for all participants were 0.164±0.038and 0.969±0.016, respectively. Contrary, average NRMSEand CORR of the modified Keir’s method had higher NRMSEand lower CORR than those of the proposed algorithm, asshown in Figs. 7(a) and 7(b). This statistical differencemeans that the performance of the proposed algorithm wassuperior to that of comparison algorithms. This is becauseour algorithm treats grip strength, sEMG signals, and armposture as independent factors by using the tensor decom-position.

On the other hand, thanks to these factor spaces in thetrained multi-factor model, our algorithm could obtain aninherent factor corresponding to arbitrary arm posture.Therefore, our algorithm was able to accurately predictgrip force even in untrained postures (see Figs. 8(a) and8(b) for Case II, Figs. 9(a) and 9(b) for Case III, and Figs.10(a) and 10(b) for Case IV). In contrast, the performanceof the multiple regression model is guaranteed by usingprobe data which is acquired under the same condition asthe training condition process [37].

Fig. 11 shows the predicted force from the proposed al-gorithm and the algorithm for comparison when subject1took an experiment with untrained posture: Wrist andforearm angles are 15◦ and 45◦, respectively. As shown inthis figure, the predicted force of the proposed algorithm ismore accurate and has a similar tendency compared to the

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Figure 7. Experimental results of the proposed method for Case 1: (a) Normalized Root Mean Squared Error (NRMSE) for each subject,(b) Coefficient of Correlation (CORR) for each subject.

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Figure 8. Experimental results of the proposed method for Case 2: (a) Normalized Root Mean Squared Error (NRMSE) for each subject,(b) Coefficient of Correlation (CORR) for each subject.

measured force than that of the comparison algorithm. Be-cause the muscle’s contraction in arm depends on not onlythe magnitude of the grasping force but also arm’s posture,our algorithm showed better performance on grasping forceprediction.

Consequently, all validation results in Figs. 7-11 wasverified that the proposed algorithm could estimate grasp-ing force with respect to various arm postures. The nextsubsection discussed why our algorithms can outperformthe comparison algorithm and a few of limitations andconsiderations of our research.

5.1. Strength of Multi-factor ModelIn general, the performance of multi-variable regres-

sion, such as the modified Keir’s method, is assured thateach input variable must be independent of each other. Inother words, low multicollinearity between input variablesshould be guaranteed [37]. However, our input variableshave strong multicollinearity, because there is a correlationbetween sEMG signals. Table 2 shows the VIF (VarianceInflation Factor, an indicator for multicollinearity) of theinput variable for each participant. As shown in Table.2, the VIF value indicated that EMG signals and arm’sangles have strong multicollinearity (In statistics, if theVIF is greater than 5, it is not suitable for the multipleregression model [37]). Hence, the modified Keir’s methodshowed low performance even though Case I validation.

Contrary, the proposed algorithm is free from this prob-lem because they perform dimension reduction on inputvariables in the process of training the multi-factor model.Thus, in Case I, the performance of the proposed algorithmshowed accurate grasping force estimation, as shown inFig. 7. In addition, in Case II-IV, our algorithm showedlow NRMSE and high CORR (see Figs. 8-10). This isthe strong evidence that our tensor decomposition basedapproach can treat each factor representing sEMG sig-nals and arm postures independently. While previous re-searches about grasping force estimation trained the re-gression model in a two-dimensional (matrix) dimension,our algorithm separates the sEMG and arm postures fromeach other, so that the correlation between the sEMG sig-nal and arm postures can be more accurately represented.

5.2. Effects of Factor InterpolationSince our algorithm finds the interpolated factors with

respect to untrained posture as mentioned in Sec. 2.3.1,it guarantees a more accurate force estimation than othercomparison algorithms. However, as shown in Figs. 8-10,the results of Case II, III, and IV by the proposed algo-rithm have slightly higher NRMSE than that of Case I. Itis because the relationship between the pattern of sEMGsignals and various posture is definitely nonlinear. To en-hance the performance of the proposed algorithm, we can

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Figure 9. Experimental results of the proposed method for Case 3: (a) Normalized Root Mean Squared Error (NRMSE) for each subject,(b) Coefficient of Correlation (CORR) for each subject.

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Figure 10. Experimental results of the proposed method for Case 4: (a) Normalized Root Mean Squared Error (NRMSE) for each subject,(b) Coefficient of Correlation (CORR) for each subject.

consider a nonlinear variable model such as Latent Vari-able Model (LVM) rather than factor interpolation pro-posed in Sec. 2.3.1 [26]. However, a more dense data setis needed to train LVM.

5.3. Computation TimeIn general, Tucker decomposition has heavier compu-

tational complexity than the general regression method inthe matrix dimension. In our validation, average compu-tation time for training the multi-factor model was 232.8sec, while that of the modified Keir’s method was 6.9 sec.However, the computation time of grasping force estima-tion based on the trained multi-factor model is sufficientlyshort enough to be available in real time. The averagecomputation time for estimating grasping force was 0.0013sec.

5.4. Limitations and ConsiderationsAlthough the proposed algorithm shows accurate grasp-

ing force estimation, we will overcome the following lim-itations to improve the performance of the proposed al-gorithm further. First of all, the proposed method onlyensures the estimation of grasping forces in a static pos-ture. Thus, if the experiment is performed while the armmoves freely, an error due to muscle contraction may oc-cur. Therefore, a study on the gripping force prediction inthe dynamic posture should be researched.

In addition, many literatures reported that the rela-tionship between grasping force and sEMG signals shouldbe changed many factors including not only arm posturebut also grasping type (i.e. precision and power grasping),grasping force pattern, and speed [38, 39]. Thus, it willbe a worthwhile study to improve the proposed algorithmin order to reflect not only arm posture but also abovefactors.

Finally, our model is based on the linear factorizationof the tensor algebra. As aforementioned in Sec. 5.2, therelationship between the grasping force and sEMG signalsis nonlinear. Thus, a nonlinear multi-factor model will beneeded to enhance the performance of the grasping forceestimator.

6. Conclusions

In this paper, a novel algorithm for predicting the grasp-ing force with respect to sEMG signals and grasping pos-ture is proposed. Unlike previous researches for graspingforce estimation, our algorithm is based on tensor algebra,so our algorithm can predict grasping forces accurately byusing the independent factors of sEMG signals and wristand forearm postures.

The most important contributions of this paper arepresented as follows. With the multi-factor model, we ob-tained the relationship between sEMG signals and grip

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(a) (b)

Figure 11. Experimental results when the subject1 takes untrained posture (wrist and forearm angles are 15◦ and 45◦): (a) estimatedgrasping force by the proposed algorithm, (b) estimated grasping force by modified Keir’s method.

Table 2. Variance inflation factor of each subject. In the table, FCR to EDC indicates sEMG signals (ym,j) in (14), respectively. In addition,Wrist and Forearm indicate the wrist’s angle (pw) and forearm’s angle (pf ). Bold font indicates the VIF is greater than 5.

Subject FCR FCU FDS ECR ECU EDC Wrist ForearmS1 5.3403 6.0247 3.2362 3.0445 5.4391 6.3275 2.5039 2.0492S2 3.3022 2.6089 3.0613 5.6034 1.8285 1.5131 1.5435 2.5977S3 6.6961 4.6880 3.4777 6.4816 6.0304 2.7443 1.1345 2.1857S4 4.8149 1.3753 2.4783 2.0484 3.5738 1.4376 1.4376 2.1132S5 7.9502 4.3281 6.5850 3.7311 3.2592 5.8507 1.9749 2.8187S6 3.0228 5.8642 2.8845 5.7069 2.8119 5.0014 1.8339 1.2075S7 4.3830 4.2515 3.0481 4.9694 3.0521 4.3959 2.2588 1.4897S8 2.8247 3.4676 5.5601 2.9593 3.0223 3.9822 1.4090 1.6302S9 2.7921 3.9240 2.9980 5.5123 3.3868 3.5156 2.7894 1.4172S10 3.5640 4.8484 4.6284 3.3893 3.2838 3.2838 1.3077 1.3499

force in various arm postures. Thanks to this multi-factormodel, the factors of untrained arm posture could be cal-culated by interpolating the existed factors of trained armpostures in the multi-factor model. Second, we showedthat grasping force could be affected by not only sEMGsignals but also various grasping posture by conductingexperiments. Finally and most importantly, we demon-strated the performance of the proposed algorithm throughvarious experiments with ten healthy people. All the sourcecodes used in the experiments are available online. In allexperiments, the NRMSE and CORR of the proposed al-gorithm are better than those of the multivariate linear re-gression algorithm. Thus, we believe that our study has agreat potential to be used in various bionic fields includingrehabilitation, human-computer interface, and myoelectriccontrol for prostheses.

Future works will involve studying a nonlinear multi-factor model to enhance the accuracy of prediction as men-tioned in Section 5.4, developing a hybrid motion/forcemapping method for controlling a prosthetic hand or robotic

hand using the proposed algorithm.

Acknowledgment

We would like to thank Sucheol Lee who designed theexperimental kit for maintaining the arm gesture. Also,we would like to thank Keunwoo Jang and Soohan Parkfor helpful comments and deep discussion.

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